A sparse generic matrix is a matrix whose entries are distinct variables and zeros. Such matrices were studied by Giusti and Merle who computed some invariants of their ideals of maximal minors. In this paper we extend these results by computing a minimal free resolution for all such sparse determinantal ideals. We do so by introducing a technique for pruning minimal free resolutions when a subset of the variables is set to zero. Our technique correctly computes a minimal free resolution in two cases of interest: resolutions of monomial ideals, and ideals resolved by the Eagon-Northcott Complex. As a consequence we can show that sparse determinantal ideals have a linear resolution over Z, and that the projective dimension depends only on the number of columns of the matrix that are identically zero. We show this resolution is a direct summand of an Eagon-Northcott complex. Finally, we show that all such ideals have the property that regardless of the term order chosen, the Betti numbers of the ideal and its initial ideal are the same. In particular the nonzero generators of these ideals form a universal Gröbner basis.

Given a linear space L in affine space A n , we study its closure L in the product of projective lines (P 1 ) n . We show that the degree, multigraded Betti numbers, defining equations, and universal Gröbner basis of its defining ideal I( L) are all combinatorially determined by the matroid M of L. We also prove I( L) and all of its initial ideals are Cohen-Macaulay with the same Betti numbers, and can be used to compute the h-vector of M . This variety L also gives rise to two new objects with interesting properties: the cocircuit polytope and the external activity complex of a matroid.

Let I be a toric ideal. We say I is robust if its universal Gröbner basis is a minimal generating set. We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely characterize all graphs which give rise to robust ideals. Our characterization shows that robustness can be determined solely in terms of graphtheoretic conditions on the set of circuits of G.Theorem 1.2. I G is robust if and only if the following conditions are satisfied.R1: No circuit of G has an even chord, R2: No circuit of G has a bridge, R3: No circuit of G contains an effective crossing, and R4: No circuit of G shares exactly one edge (and no other vertices) with another circuit such that the shared edge is part of a cyclic block in both circuits.

A Lissajous knot is one that can be parameterized as K(t) = (cos(nxt + φx), cos(nyt + φy), cos(nzt + φz))where the frequencies nx, ny, and nz are relatively prime integers and the phase shifts φx, φy and φz are real numbers. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems which allow us to place bounds on the number of Lissajous knot types with given frequencies and to efficiently sample all possible Lissajous knots with a given set of frequencies. In particular, we systematically tabulate all Lissajous knots with small frequencies and as a result substantially enlarge the tables of known Lissajous knots.A Fourier-(i, j, k) knot is similar to a Lissajous knot except that the x, y and z coordinates are now each described by a sum of i, j and k cosine functions respectively. According to Lamm, every knot is a Fourier-(1, 1, k) knot for some k. By randomly searching the set of Fourier-(1, 1, 2) knots we find that all 2-bridge knots up to 14 crossings are either Lissajous or Fourier-(1, 1, 2) knots. We show that all twist knots are Fourier-(1, 1, 2) knots and give evidence suggesting that all torus knots are Fourier-(1, 1, 2) knots.As a result of our computer search, several knots with relatively small crossing numbers are identified as potential counterexamples to interesting conjectures. * Supported by NSF REU grant DMS-0453284.

We call an ideal in a polynomial ring robust if it can be minimally generated by a universal Gröbner basis. In this paper we show that robust toric ideals generated by quadrics are essentially determinantal. We then discuss two possible generalizations to higher degree, providing a tight classification for determinantal ideals, and a counterexample to a natural extension for Lawrence ideals. We close with a discussion of robustness of higher Betti numbers.